A three-digit number increases by nine if you exchange the second and third digits. The same three-digit number increases by if you exchange the first and second digits. By how much will the number increase if you exchange the first and third digits of the same number?Source: mathcontest.olemiss.edu 11/14/2011

SOLUTIONExchange second and third digits
Suppose the three-digit number is . If we exchange the second and third digits, the resulting number is . The increase is much greater than 9.
So we try a smaller number for example and exchange the second and third digits to obtain . The increase equals which is the correct amount. But, this is true also of , etc.

Exchange first and second digits
From what we learned above the first digit should be 1 less than the second digit if we want the increase to equal . For example, if we start with the three-digit number and exchange the first and second digits to obtain , the increase equals as expected. But, this is true of the many more numbers as shown below.

123

213

213 – 123 = 90

234

324

324 – 234 = 90

345

435

435 – 345 = 90

456

546

546 – 456 = 90

567

657

657 – 567 = 90

678

768

768 – 678 = 90

789

879

879 – 789 = 90

Exchange first and third digitsIf we exchange the first and third digits, the increase equals 198.